Table Of ContentExtracting high fidelity quantum computer
hardware from random systems
0
1 A. Walther, B. Julsgaard , L. Rippe , Yan Ying, S. Kr¨oll
0 ‡ §
2 Department of Physics, Lund University, P.O. Box 118, SE-22100Lund, Sweden
n
a
J R. Fisher, S. Glaser
1
Department Chemie, Technische Universit¨at Mu¨nchen, Lichtenbergstrasse 4, 85747
1
Garching, Germany
]
h E-mail: [email protected]
p
-
t Abstract. An overview of current status and prospects of the development of
n
a quantum computer hardware based on inorganic crystals doped with rare earth ions
u is presented. Major parts of the experimental work in this area has been done in two
q
places, Canberra, Australia and Lund, Sweden, and the present description follows
[
more closely the Lund work. Techniques will be described that include optimal
1
filtering of the initially inhomogeneously broadened profile down to well separated
v
andnarrowensembles,aswellastheuseofadvancedpulse-shapinginordertoachieve
4
6 robust arbitrarysingle-qubit operations with fidelities above 90%, as characterizedby
6 quantum state tomography. It is expected that full scalability of these systems will
1
requiretheabilitytodeterminethestateofsinglerare-earthions. Ithasbeenproposed
.
1
that this can be done using special readoutions doped into the crystaland an update
0
is given on the work to find and characterize such ions. Finally, a few aspects on
0
1 possibilities for remote entanglement of ions in separate rare-earth-ion-dopedcrystals
: are considered.
v
i
X
r
a PACS numbers: 3.65.Wj, 03.67.Hk, 42.50.Md
1. Introduction
Manysolidstatequantumcomputingsystemsrelyonatopdownapproachwithcarefully
crafted systems, where parameters are designed during the manufacturing or growth
process, for instance, superconducting qubits [1, 2], quantum dots [3] and the Kane
model [4]. At the other end of the scale there are bottom up approaches, where
useful hardware can be extracted out of random materials which intrinsically have very
favorableproperties, such asnitrogenvacancy (NV)-centers[5] andrare-earth-ion-doped
inorganic crystals [6]. In this paper we will discuss development of quantum computer
Now with Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120,DK-8000,
‡
Aarhus C
Now with National Center for Atmospheric Research, NCAR, Boulder, CO
§
Extracting high fidelity quantum computer hardware from random systems 2
hardware based on inorganic crystals doped with rare earth ions. Such crystals are
used extensively in current technology, e.g. as active laser materials or as scintillating
materials in X-ray detectors. They are manufactured in single crystal growth processes
where a dopant material is added during the growth such that the dopant ions replaces
host ions in the regular crystal lattice at concentrations that may range between 0.01%
and 10% depending on application. Although the average concentration is defined
by the dopant concentration in the crystal melt, the relative positions of the dopant
ions is largely random. However, randomness and less precise control of construction
and growth parameters can be compensated for by using suitable quantum computing
schemes and by developing robust pulses enabling high fidelity qubit operations.
The next section (Section 2) will describe basic properties of rare-earth-ion-doped
crystals relevant for quantum information. This includes coherence times which can
be several seconds [7] and ion-ion interactions as a mechanism for gate operations.
For a laser beam passing through a rare-earth-ion-doped crystal there may be 1015
ions within the laser focal region. However, qubits can still be created and prepared
in well defined states and arbitrary qubit operations, characterized by full quantum
state tomography, give fidelities, F, in the range 0.91 < F < 0.96 [8]. A brief
description on the experimental setup, including the pulse-shaping system, will be given
in Section 3, while Section 4 describes how to efficiently initialize the system into well
defined qubits, which are inhomogeneous subgroups out of these 1015 ions. In Section 5
we show experimental data on qubits prepared in arbitrary superposition states and
characterized by quantum state tomography. The high fidelity results are all based on
qubit operations that are robust against individual differences between the ions [9, 10].
We will also present data from the development and use of optimal pulses sequences [11]
for operation fidelities > 0.96. Section 6 starts with a description of how to, in contrast
to the experiments mentioned above, create qubits in these crystals that each consist
of just a single ion, which offers a better scalability. The single ion qubit scheme [12]
is based on co-doping the crystals with ”readout ions”. Such readout ions should have
a short upper state lifetime in order to enable repeated cycling. One such candidate is
the 4f to 5d transition in Ce3+ and experimental data to characterize this transition is
also presented in Section 6. The last Section before the summary contains a discussion
on entanglement of remote qubits and further scalability.
2. Rare-earth-ion-doped crystals
Amajorstrengthofusingrare-earthionsasqubitsarethelongcoherence times. Inthese
ions, transitions within unfilled electron shells are partially shielded from environmental
dephasing mechanisms by energetically lower, but outer-lying, filled shells. The present
resultswereobtainedusing the3H 1 D transitioninPr3+:Y SiO at606nm[13]. For
4 2 2 5
→
temperatures below 4 K, the Pr3+ ions in this material has an electronic state lifetime of
164 µs, and a coherence time, T , of about 100 µs (somewhat depending on the density
2
of excited states in the material). By applying magnetic fields however, this time can
Extracting high fidelity quantum computer hardware from random systems 3
be increased, and there are also other rare-earth ions which have even longer coherence
times, for example Er, where a T of 6.4 ms has been measured [14].
2
The qubit states are not represented by electronically
4.8MHz e
excited states though, but rather by ground state hyperfine 4.6MHz
levels, shown in the level diagram of Pr3+ in Figure 1. For
Pr3+:Y SiO , the hyperfine lifetime is about 90 s and the (cid:90),(cid:58)
2 5 0 0 (cid:90),(cid:58)
coherence time about 500 µs. The hyperfine T can also 1 1
2
be increased by magnetic fields, and up to 860 ms has
0
been demonstrated [15]. In addition, dynamic decoupling
10.2 MHz
1
sequences (bang-bang pulses) can also be applied to further
17.3 MHz
aux
increase the coherence time to more than 30 s [7].
The rare-earth ions are randomly doped into the host
Figure 1: Relevant en-
crystals, replacing one of the host ions in the lattice.
ergy level structure for
This replacement causes changes to the crystal electric
the Pr3+ ion transition
field which then causes each rare-earth ion to pick up a
utilized in the present
shift in its resonance frequency. This amounts to a large
work.
inhomogeneous broadening of the absorption line. For
Pr3+, the inhomogeneous linewidth is about 5 GHz, which can be compared to the
homogeneous linewidth of about 3 kHz, given by the optical T , mentioned above.
2
In the rare-earth quantum computing scheme, the inhomogeneous structure is of
significant importance, as multiple qubits are addressed in the frequency domain, each
qubit having a different resonance frequency. From above, the ratio of inhomogeneous
over homogeneous linewidth is about 106 for Pr3+:Y SiO , but since ions do not have
2 5
only a single transition frequency, the number of frequency channels are in practice
much smaller. Fig. 1 shows that the lowest and highest transition frequency for a given
ion differ by 4.8+4.6+10.2+17.3 = 36.9 MHz. Thus, about 100 individual frequency
channels are a more realistic value. The Pr3+ ion sits in a non-centrosymmetric site, and
thus has a permanent electric dipole moment. The value of this dipole moment depends
on the electron configuration and the value of the dipole moment of an electronically
excited state, is therefore different from that of the ground state. Qubit-qubit gates can
then be executed through dipole-dipole interactions. Since the dipole moment changes
when the ion becomes excited, exciting an ion will shift the transition frequency of all
near-lying ions. An ion belonging to one qubit can then be controlled by the excitation
of a another ion, belonging to another qubit, if they are spatially sufficiently close to
each other. This effect is used as the control mechanism for conditional gates, and the
basis for this was demonstrated in [10].
3. Experimental setup
The experimental system is based around a Coherent 699-21 ring dye laser, stabilized to
1 kHz linewidth, by locking to a spectral hole in a Pr3+:Y SiO crystal [16]. The rare-
2 5
earth crystal used in the experiments is a 0.5 mm thick Pr3+:Y SiO crystal with a Pr3+
2 5
Extracting high fidelity quantum computer hardware from random systems 4
doping concentration of 0.05%, which gives a maximum optical density of about 2 3.
−
In order to get rid of phonon interactions, the crystal is submerged in liquid helium and
cooled to 2 K. The available laser power is in the order of 100 mW before the cryostat,
which translate into maximum Rabi frequencies of about 1-3 MHz, depending on which
transition is targeted, given a laser focus of 100 µm.
High fidelity gates requires a high-performance pulse-shaping system, as will be
further discussed in Section 5. Two acousto-optic modulators (AOMs) are used to
create the desired pulses. The incoming continuous wave (CW) photons are deflected
by an RF acoustic wave in the AOMs. By varying the power of the acoustic wave,
the amplitude of the deflected light is controlled. By virtue of energy and momentum
conservation, the deflected photons must increase their frequency by an amount equal
to that of the phonons. Thus, by varying the frequency of the acoustic wave we directly
control the frequency of the light pulse, and furthermore, by changing the phase of the
acoustic wave we also gain control over the optical phase of the light, which is necessary
for controlling the coherent interaction with the ions. For these modulators, the power
conversion from RF wave to light amplitude is quite non-linear. In order to achieve high
accuracy of the pulse amplitude envelopes over as much as six orders of magnitude in
dynamicrange, anelaboratecalibrationsystem, thatalsoincludesthepowerdependence
on RF frequency, was setup. The first AOM has a center frequency of 200 MHz and is
aligned in a double pass configuration, in order to cancel out spatial movement of the
deflected beam. After both passes the AOM has a total deflection range of 200 MHz.
The second AOM has a 360 MHz center frequency, and is used only to split a pulse at
a single frequency into two frequencies, such that an excitation pulse can be resonant
with the 0 e and 1 e transitions at the same time, as further discussed in
| i → | i | i → | i
Section 5.1.
4. Qubit initialization
Startingfromtheinhomogeneous distribution, we employ advanced optical hole-burning
sequences in order to extract a useful qubit system. Initially, the ions occupy all three
ground states, and the absorption at any given frequency within the absorption profile,
contains contributions from all nine possible transitions (see Figure 1). The goal of the
hole-burning is to initialize a narrow controlled ensemble of ions into the 0 -state, well
| i
separated in frequency from any other ion.
At first, a wide spectral hole, henceforth called a pit, is created. The hyperfine
level splittings set the limit for the maximum frequency range of such a pit. Since the
ions must be in one of the ground states, it can not be larger than 27.5 MHz, as can
be seen in Figure 1. This maximum width is then further reduced by the excited state
splitting down to 18.1 MHz. Scanned laser pulses target specific transitions inside this
18 MHz region, and are repeated in order to create a spectral pit where all ions have
been removed within an interval of maximum 18 MHz. Such a simple pit is shown in
Figure 2a.
Extracting high fidelity quantum computer hardware from random systems 5
(a) Simple pit Laser scan (b) Optimal pit
n
n o
ptio rpti
r o
o s
s b
b A
A
<18 MHz 10.2 MHz 17.3 MHz
)
u. 0 1 aux
b. 1.2 L) 2.5
ar 1 α( 2
e (0.8 on
nc0.6 pti 1.5
ce or 1
s0.4 s
e b
or0.2 A 0.5
Flu 0-10 -5 0 5 10 0-20 -10 0 10 20 30
Frequency (MHz) Frequency(MHz)
Figure 2: (color online) a) shows a simple pit created by repeated scanning across an
interval less than 18 MHz. Panel b) displays an improved pit, where scans on multiple
intervals have been iteratively repeated in order to shuffle all ions outside the pit as far
out as possible, thereby creating an optimal pit. Also note that the scales in the two
experimental figures are different.
Eventually, qubit operation pulses will be used inside the pit structure, and it then
becomes important to minimize the interactions between the tails of such pulses and
the edges of the pit. The pit shown in Figure 2a displays non-optimal edges, that are
a bit rounded and also go up to the maximum αLimmediately. In order to improve
on this, a more elaborate pit-burning scheme was introduced. In this scheme, pulses
that target specific levels of Pr3+ outside the pit are used in order to burn them even
further out, as illustrated by the top part of Figure 2b. For an ion having a particular
frequency relative to the pit, as the ion illustrated in this figure, level 0 and 1 are
| i | i
targeted in an optical pumping process that eventually transfers them to their aux
| i
state. After this, ions of other frequency ranges might fall back down inside the pit,
and the initial pit-burning pulses will have to be repeated iteratively together with the
secondary pulses to keep the center of the pit empty. A similar procedure is then also
repeated on the left side of the pit in order to achieve an optimal result, shown in the
lower part of panel b. A more extensive description of the burning procedure can be
found in [17] and a detailed list of the exact pulses used to burn such an optimal pit
can be found in [18].
After an empty spectral pit has been created, a qubit ensemble can be initialized
insidethepitbya”burnback” pulsesequence. Thesepulsescanwithbenefitsbecoherent
pulses that transfer ions from outside the pit, say at higher frequencies, first to the
Extracting high fidelity quantum computer hardware from random systems 6
4.8MHz
(a) 4.6MHz
0.6
1
1 2 3
0.4
) 2
L 10.2 MHz
α 0.2 3
n ( 17.3 MHz
o 0
orpti0.6 (b) 5 4.8MHz
s 4.6MHz
b0.4
A 4
4 5
0.2
10.2 MHz
0 17.3 MHz
-2 0 2 4 6 8 10 12 14 16 18
Frequency(MHz)
Figure 3: (color online) a) shows the rare-earth qubit system initialized to the 0 state,
| i
visible through the three absorption peaks from this state. Panel b) then shows the
system after a coherent transfer over to the 1 state.
| i
excited state, and from there down to one of the ground states in a controlled manner.
Figure 3a shows the system initialized to one of the ground states, in this case the 0
| i
state.
5. Qubit gate operations
The ensemble qubit is now initialized to the 0 state. In order to be able to
| i
perform high fidelity gate operations, we must now find pulses that compensate for
the inhomogeneous broadening of the ensemble, which would otherwise cause dephasing
during the excitation. One type of pulses that are robust against such dephasing are the
complex hyperbolic secant pulses, or sechyp for short. The amplitude envelope of these
followahyperbolicsecant shape, whileatthesametimethepulseischirpedaccordingto
a hyperbolic tangent function, centered around the mean ensemble transition frequency.
Such pulses can efficiently transfer an inhomogeneous ensemble between the two poles
on a Bloch sphere, as illustrated in Figure 4a. The different lines correspond to ions
with different atomicresonance frequencies, which canall be seen to converge at thetop.
The robustness of these pulses has been verified experimentally by transfer efficiencies
of 97% for ensembles inhomogeneously broadened more than 100 linewidths. Figure 3b
shows the system after such a sechyp transfer from 0 e immediately followed by
| i → | i
a transfer from e 1 . In addition to this, the sechyp pulses also have the good
| i → | i
property that they are robust against fluctuations in laser power. While normal square-
or gaussian-shaped pulses need to have very well tuned values of laser intensity and
duration in order to get the correct pulse area, the sechyp pulses simply need to be
above a certain threshold Rabi frequency, above which they always work with high
efficiency. This independence of the light intensity can be intuitively perceived from
the Bloch sphere in Figure 4a, where we see that as the different components approach
Extracting high fidelity quantum computer hardware from random systems 7
(a) (b) z (c) z
e
z 0
θ
1
2
D
B
y y y
x x
x
1 B
Figure 4: (color online) a) shows the paths taken by three ions with different resonance
frequencies, while being transferred to the excited state by a coherent sechyp pulse.
Panel b) illustrates the dark and the bright states on the Bloch sphere, relevant for
the gate operations, and panel c) shows a full qubit gate operation by means of two
consecutive dark state pulses (two-color pulses). Note that the transfer in c) is going
from and to the bright state.
their final state, they circle around the top pole, and a higher Rabi frequency would just
make them circle around the z-axis faster, not take them away from the desired state.
These pulses were investigated more carefully in [9].
5.1. Dark state gates
The sechyp pulse-shapes solve the problem of inhomogeneous dephasing as well as offer
a robustness with respect to fluctuating experimental parameters, but can only perform
pole-to-pole transfers, not create arbitrary superposition states. The qubit is the two
hyperfine levels denoted 0 and 1 in Figure 1, and to perform operations between
| i | i
these states, two-color fields, which are resonant with 0 e and 1 e at the
| i → | i | i → | i
same time, are employed [9]. Determined by the phase relation, φ, between the two
fields, two superposition states, shown in Figure 4b, the bright and dark state:
B = 1 0 e iφ 1
| i √2 | i− − | i (1)
D = 1 (cid:16) 0 +e iφ 1 (cid:17)
| i √2 | i − | i
(cid:16) (cid:17)
can be defined. Similarly to electromagnetically induced transparency (EIT), the dark
state does not interact with the light. Thus, the two-color pulse will only target the
brightstate, andtwoconsecutivetransfers, firstfrom B e andthenfrom e B ,
| i → | i | i → | i
but going down along a different path as illustrated in Figure 4c, will add a phase angle
enclosed by the rotation, θ, to the bright state: B eiθ B . This operation can be
| i → | i
rewritten in the computational basis ( 0 , 1 ), and then becomes
| i | i
cos(θ/2) ieiφsin(θ/2)
U = eiθ/2 . (2)
dark " ie−iφsin(θ/2) cos(θ/2) #
Extracting high fidelity quantum computer hardware from random systems 8
(a) (b)
0.6 X 0 1 z
0.4
0.2
) 0
L
α( 0.6 Y
n 0 1
o 0.4
pti
or 0.2 y
s
b 0
A
0.6 x
Z
0 1
0.4
0.2
0
0 5 10 15
Frequency (MHz)
Figure 5: (color online) a) shows the projective measurements on the x-, y- and z-axis
respectively. The two left peaks are both transitions from the 0 state and correspond
| i
to a +1 measurement value, while the two right peaks similarly are transitions from the
1 state and correspond to a 1 value with the respect to the current axis. Panel b)
| i −
shows the result of a quantum state tomography of the six states along the positive and
negative axes.
which is a matrix describing an arbitrary rotation around any vector on the equator of
the Bloch sphere. By picking suitable phase angles, one can thus perform any qubit
gate operation [9].
These dark state gates were implemented and the qubit superposition states that
were created as a result of the rotations, were characterized using quantum state
tomography. In Figure 5a the readout of a particular superposition state, 1 ( 0 1 )
√2 | i−| i
is shown as an example. The three curves correspond to projection measurements on
the x-, y- and z-axis respectively. As expected, we see that both the y and the z
measurement contain equal contributions from either state, but the x measurement
gives a projection onto one of the states. States corresponding to the six different
(positive and negative) axes on the Bloch sphere, were prepared, and characterized, and
the resulting state vectors inscribed in a Bloch sphere are displayed in Figure 5b.
The z-projection was done through an absorption measurement, where the two
leftmost peaks in Figure 5 correspond to transitions from the 0 state and the two
| i
rightmost peaks correspond to transitions from the 1 state. In order to do the x- and
| i
y-measurements, suitable rotations were first performed to rotate those axes onto the
z-axis, such that an absorption measurement now gives the same result as a direct x- or
y-measurement would have, in the original basis. From those measurements the density
Extracting high fidelity quantum computer hardware from random systems 9
matrix can then be calculated through (see e.g. [19])
tr(ρ)I +tr(Xρ)X +tr(Yρ)Y +tr(Zρ)Z
ρ = , (3)
2
where X, Y and Z correspond to the Pauli matrices, which together with the identity,
I, span the 2 2 qubit space in the density matrix representation. The above equation
×
is useful because the traces correspond to actual physical measurements. For example,
tr(Zρ) means measuring the projection of the unknown state onto the Z axis, which
will yield a value between -1 and +1, and the same for the other axes. The fidelity of
the entire procedure can then be calculated from
F = ψ ρ ψ , (4)
tot theor exp theor
h | | i
i.e. the overlap between the theoretical and the experimental representations. For the
six states displayed in Figure 5b, the total fidelities were all between 0.84 and 0.92. As
mentioned above the total procedure consists of two rotations, first one to create the
state, then another oneto performthe measurement inthe right basis. Thus, thefidelity
for a single qubit gate can be expressed as F = √F , and we see that the qubit
gate tot
gate fidelities are in the range of 0.91 < F < 0.96. More information and discussion
gate
regarding these results can be found in [8, 18].
5.2. Optimal control pulses
Even though the dark state qubit gates were realized with good fidelity there are still
reasons to search for further improvements. The total duration of the dark state pulses
that performs one arbitrary qubit rotation is close to 18 µs. The employed laser power
on the other hand, typically provides Rabi frequencies in the order of 1 MHz, which
indicates that the qubit rotations could probably be performed an order of magnitude
faster, with an optimal scheme. To reduce the operation time, pulses obtained through
optimal control theory calculations are currently investigated.
Optimal control theory provides a framework for finding the best set of controls to
steer a system so that a desired target state or unitary gate is implemented. In some
special cases these controls can be determined analytically, see for example [20, 21, 22].
In addition, powerful numerical methods [23] are available that make it possible to
explore the physical limits of time-optimal control experiments in cases where no
analytical solutions are known. These methods have so far been successfully applied
to spin systems [24, 25, 26] and superconducting qubits [27], and trapped ions [28].
Optimized pulses can be designed to account for experimental errors, and, if a realistic
model is available, can include multiple system levels and transitions.
The basic idea of optimal control algorithms is to minimize a cost function, which is
determinedbythephysical system athand. Inthepresent case, thisisgivenbythelight-
field interaction with the Pr3+ ion, and in addition to this, a number of experimental
limitations, such as AOM maximum bandwidth or laser power fluctuations, are enforced
aswell. The algorithmthen finds apulse shape that performsa particular operationtoa
specified minimumfidelity, whilestayingwithintherestrictions. Itcanbenotedthatthe
Extracting high fidelity quantum computer hardware from random systems 10
(a) amplitude envelope (b) phase
10
1
0.8 5
0.6
e
d 0.4 0
u
mplit 00.2 -5
a -0.2
ed 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 -109 10 11 12 13 14
z
ali 10
m 1
or 0.8 5
n
0.6
0.4 0
0.2
-5
0
-0.2
-10
9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 9 10 11 12 13 14
time (µs) time (µs)
Figure 6: (color online) Two pulse shapes with different bandwidths, both designed by
optimal control algorithms to perform efficient state-to-state transfers. The left side
shows the detected beating and the amplitude envelope, while the right side shows the
phase evolution. In both cases the red line is the theoretically desired shape, while the
blue line is the experimentally detected shape, and in both cases they are matching very
well. Note that in the phase plot, the phase does not have any meaning outside the
pulse envelope (since the amplitude there is zero).
solution is not necessarily unique, or even at the global maximum, and the shapes that
comes out of the calculations are then often strange-looking, without obvious intuitive
explanations. In order to characterize how well the AOM system could reproduce these
strange shapes, an interference experiment was set up, where the AOM pulse-shaping
system is put in one arm and the light passing through this part then interferes with
the light from an undisturbed part of the laser. The result is a beating, from which
both the amplitude envelope and the phase evolution of the pulses can be obtained via
a Fourier transform. Figure 6 shows two examples of such pulses. In panel a) of this
figure, the green, oscillating curve is the detected beating, the blue line is the amplitude
envelope of the beating, which is almost perfectly overlapping with the red line, which
is the theoretically desired shape (may be hard to see if not in color).
The upper and lower pulse differs in allowed bandwidth. The lower pulse has a
bandwidth of about 16 MHz, while the upper pulse has been restricted to work with
a bandwidth of only 2 MHz. Ideally, one would want as high bandwidth as possible,
since this enables pulses with shorter duration. The right side of Figure 6 displays the
phase evolution of the detected pulses as well as the desired phase chirp, and again,
there is almost perfect overlap. In order to test the optimal control results, a series
of five pulses were designed, each one with the same objective, to do a state-to-state
transfer from 0 to e , but with different bandwidths. The resulting efficiency as a
| i | i
function of bandwidth is shown in Figure 7. As can be seen, the efficiency is very good
